Introduction

The purpose of this library is to provide a method for evaluating both transient and steady-state M/M/C/K single queues and queueing networks. The library has been designed to make this process as smooth as possible by restricting the input to the adjacency matrix and the basic parameters that characterize the system.

Using the library involves the following two steps:

Step 1: Construct the infinitesimal generator for the network at hand. This is handled automatically by the create class.

Step 2: Use the object created in the aforementioned step to retrieve the behavior of the network. This is handled by the evaluate class.

Basic Overview

Files

• mc_math-1.2.jar: The library file.

Classes

• create: Automatically constructs the infinitesimal generator (the transition rate matrix) and various other parameters.

• evaluate: Uses an object defined with create to evaluate the queueing network.

Getting Started

Firstly, download and add mc_math.jar to your Java-project.

Now import the queueing classes.

import queueing.*;

Define the weighted directed adjacency matrix using the structure: (1) Sources nodes, (2) queueing nodes, and (3) sink node. In this example, we have three queueing nodes. A single source node feeds all arriving customers into the first queueing node. The flow then splits into three parts sending 45% of the customers to queue 2, 30% to queue 3, and 25% to the sink node. Queue 2 and 3 send all customers to the sink after their service has been completed.

double[][] A = {{0,1,0,0,0},
                {0,0,0.45,0.30,0.25},
                {0,0,0,0,1},
                {0,0,0,0,1},
                {0,0,0,0,0}};

Define the remaining characteristics of the system, i.e. the arrival rate (lambda), service rates (mu), number of servers (c), capacity (cap), and how much of the capacity is occupied at time=0 (occupiedCap). If customers should be rejected when downstream queues are full, set rejectWhenFull = true; otherwise rejectWhenFull = false.

double[] lambda = {2}; double[] mu = {1.5,4,2.5}; int[] c = {2,1,2}; int[] cap = {20,20,20};
int[] occupiedCap = {0,0,0}; boolean rejectWhenFull = false;

Create the model.

create network = new create(A,lambda,mu,c,cap,rejectWhenFull);

Prepare the evaluation calculations by plugging the network object into evaluate.

evaluate system = new evaluate(occupiedCap,network);

Evaluate the behavior of the system at time=5 with a precision of 1x10^-9.

system.uniformization(5,1e-9);

Evaluate the steady-state behavior of the system (i.e. at time=Inf) with a precision of 1x10^-6.

system.gauss_seidel(1e-6);

Get the marginal state distribution for each of the network queues.

double[][] dist = system.getMarginalDistributions();

Get the expected number of customers within each queue node.

double[] expValue = system.expectedCustomers();

Get the probability of waiting on arrival at each queue node.

double[] waitProb = system.waitingProbability();

Citing this library

Anders Reenberg Andersen. (2021). areenberg/mc_math: Converted to Maven project (v1.2). Zenodo. https://doi.org/10.5281/zenodo.5650104

License

Copyright 2021 Anders Reenberg Andersen, PhD

Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.